Antiderivatives

An antiderivative of a function
f
is a function whose derivative is
f
. In other words,
F
is an antiderivative of
f
if
F' = f
. To find
an antiderivative for a function
f
, we can often reverse the process of
differentiation.

For example, if
f = x4
, then an antiderivative of
f
is
F = x5
, which can be found by reversing the power rule.
Notice that not only is
x5
an antiderivative of
f
, but
so are
x5 + 4
,
x5 + 6
, etc. In fact, adding or
subtracting any constant would be acceptable.

This should make sense algebraically, since the process of taking the
derivative (i.e. going from
F
to
f
) eliminates the constant term of
F
.

Because a single continuous function has
infinitely many antiderivatives, we do not refer to "the antiderivative",
but rather, a "family" of antiderivatives, each of which differs by a
constant. So, if
F
is an antiderivative of
f
, then
G = F + c
is also
an antiderivative of
f
, and
F
and
G
are in the same family of
antiderivatives.

Indefinite Integral

The notation used to refer to antiderivatives is the indefinite integral.
f (x)dx
means the antiderivative of
f
with respect to
x
. If
F
is an antiderivative of
f
, we can write
f (x)dx = F + c
. In this context,
c
is
called the constant of integration.

To find antiderivatives of basic functions, the following rules can be used:

xndx = xn+1 + c
as long as
n
does not equal -1.
This is essentially the power rule for derivatives in reverse

cf (x)dx = cf (x)dx
.
That is, a scalar can be pulled out of the integral.

(f (x) + g(x))dx = f (x)dx + g(x)dx
.
The antiderivative of a sum is the sum of the antiderivatives.